How to turn off what seems to be an auto extrude
Blender has no auto extrude functionality like what you describe to my knowledge. Based on your description, the most likely culprits are.


Duplicate vertices, to fix, Tab into Edit mode and perform a Remove Doubles operation from the W specials menu.
Proportional Editing, this is usually indicated by a circle that shows the region being affected, check your view header for a blue icon/toggle this with O. See Why is an object stuck to another?


Or it could be a combination of both of the above.